HMMT 十一月 2023 · GEN 赛 · 第 5 题

HMMT November 2023 — GEN Round — Problem 5

On an 8 × 8 chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of k . (Two rooks attack each other if they are in the same row or column and no rooks are between them.)

解析

On an 8 × 8 chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of k . (Two rooks attack each other if they are in the same row or column and no rooks are between them.) Proposed by: Arul Kolla Answer: 14 Solution: The answer is k = 14 . For a valid construction, place the black rooks on cells ( a, a ) for 2 ≤ a ≤ 7 and the white rooks on cells ( a, a + 1) and ( a + 1 , a ) for 1 ≤ a ≤ 7 . R R r R R r R R r R R r R R r R R r R R Now, we prove the optimality. As rooks can only attack opposite color rooks, the color of rooks in each row is alternating. The difference between the number of black and white rooks is thus at most the number of rooks. Thus, k ≤ 6 + 8 = 14 .